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Lecture 2

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Correction Stockinger, - SUSY skript, http://iktp.tu-dresden.de/Lehre/SS2010/SUSY/inhalt/SUSYSkript2010.pdf Drees, Godbole, Roy - "Theory and Phenomenology of Sparticles" - World Scientific, 2004 Baer, Tata - "Weak Scale Supersymmetry" - Cambridge University Press, 2006 Aitchison - "Supersymmetry in Particle Physics. An Elementary Introduction" - Institute of Physics Publishing, Bristol and Philadelphia, 2007 Martin -"A Supersymmetry Primer" hep-ph/9709356 http://zippy.physics.niu.edu/primer.html Unfairly criticised: Now included full superfield chapter (as of 06/09/2011)

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First lets review what we learned from lecture 1…

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1.2 SUSY Algebra (N=1) From the Haag, Lopuszanski and Sohnius extension of the Coleman-Mandula theorem we need to introduce fermionic operators as part of a “graded Lie algebra” or “superalgerba” introduce spinor operators and Weyl representation: Note Q is Majorana (Recap of Lecture 1)

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Weyl representation: Immediate consequences of SUSY algebra: ) superpartners must have the same mass (unless SUSY is broken). Non-observation ) SUSY breaking (much) Later we will see how superpartner masses are split by (soft) SUSY breaking (Recap of Lecture 1)

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Weyl representation: (Recap of Lecture 1) Already saw significant consequences of this SUSY algebra: OR

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Weyl representation: (Recap of Part 1) Already saw significant consequences of this SUSY algebra: decreases spin

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Extension of electron to SUSY theory, 2 superpartners with spin 0 to electron states We have the states: Electron spin 0 superpartners dubbed ‘selectrons’ The spins of the new states given by the SUSY algebra SUSY chiral supermultiplet with electron + selectron: Take an electron, with m= 0 (good approximation): Simple case (not general solution) for illustration

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Lecture 2

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Supersymmetry is a symmetry of the S-matrix. So, So SUSY gives relations between processes involving the pariticles and those with their superpartners. ) Very predictive. SUSY cross-sections 4E

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Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom Proof: Witten index

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Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom Proof: Witten index

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Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom swap Proof: Witten index

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Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom swap Proof: Witten index

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Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom swap Proof: Witten index

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Degrees of freedom In SUSY: number of fermionic degrees of freedom = number of bosonic degrees of freedom swap Proof: Where we have used completeness of the set,, twice on the second term in lines 2 & 3 Note: proof assumes and may not be true in the ground state if SUSY is unbroken Witten index

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Weyl representation: Recall SUSY algebra lead to: 2 states from SM fermion: 2 bosonic states Electron spin 0 superpartners dubbed ‘selectrons’

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Superpartners Analogously for a scalar boson, e.g. the Higgs, h, has a fermion partner state with either and a gauge boson with s = 1, -1, has a partner majorana fermion as superpartner Higgs, h, with Higgsino with Fermions Sfermions with Vector bosons Gauginos with Warning: Hand waving (details later)

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2. SUSY Lagrange density How do we write down the most general SUSY invariant Lagrangian? – construct using two component Weyl spinors, by examining the transformations of scalars, fermions and gauge boson Brute force (See Steve Martin’s primer or Aitchison)* superfields/ superspace – work in a simpler formalism which treats the supersymmetry as an extension of spacetime and superpartners as components of a superfield. (Drees et al, Baer & Tata, our lectures) *Martin now has a full chapter on superfields where he contructs the Lagrangian in a similar way to us, but maintains the brute force approach in earlier chapters

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2.2 Superspace Lorentz transformations act on Minkowski space-time: In supersymmetric extensions of Minkowki space-time, SUSY transformations act on a superspace: 8 coordinates, 4 space time, 4 fermionic Grassmann numbers

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Notational aside: 4 –component Dirac spinors to 2-component Weyl spinors Dirac spinor 2 component Weyl spinors Under Lorentz transformation Form representaions of lorentz group and Left handed Weyl spinor Right handed Weyl spinor

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Notational aside: 4 –component Dirac spinors to 2-component Weyl spinors Under Lorentz transformation Form representaions of lorentz group and 2 component Weyl spinors Right handed spinor Left handed spinor

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Dirac spinor2 component Weyl spinors We define: Note Bilinears Lorentz scalar Warning: take care with signs!

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Bilinears Lorentz scalar Dirac spinor2 component Weyl spinors Warning: take care with signs!

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Bilinears Lorentz scalar Dirac spinor2 component Weyl spinors Warning: take care with signs!

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Bilinears Lorentz scalar Dirac spinor2 component Weyl spinors Warning: take care with signs! Home Exercise: prove identities! Further Identities

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Dirac spinor2 component Weyl spinors Right handed spinor Left handed spinor

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Dirac spinor2 component Weyl spinors Right handed spinor Left handed spinor

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Dirac spinor 2 component Weyl spinors Right handed spinor Left handed spinor

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Dirac spinor2 component Weyl spinors Right handed spinor Left handed spinor For Majorana spinor:

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Grassmann Numbers Anti-commuting “c-numbers” {complex numbers } If{Grassmann numbers} then Similarly Differentiation: Integration:

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